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Using Properties to Make Addition Easier

Date: 10/15/2003 at 21:30:37
From: Linda
Subject: adding mentally

My daugher's homework is asking her to "add mentally, using the
commutative, associative, and identity properties to help you add."

Here's one example:

     6    --
     3    --
     0    --
     2    --
     4    --
   + 5  + --
   ---  ----                        

She doesn't know what they are asking her to do.  


Date: 10/16/2003 at 01:20:26
From: Doctor Ian
Subject: Re: adding mentally

Hi Linda, 

Let me illustrate how I would use these properties to add these numbers.  

  6 + 3 + 0 + 2 + 4 + 5

The commutative property of addition says that when we add numbers, it
doesn't matter what order we add them in.  That means we can move the
numbers around so that convenient pairs are next to each other.  I'd
want to move the 4 next to the 6, and the 2 next to the 3:

  6 + 4 + 3 + 2 + 0 + 5

Why would I want to do this?  Because 6 and 4 add up to 10, which is
easy to deal with:

  10 + 3 + 2 + 0 + 5

Also, 3 and 2 add up to 5, which is easy to deal with, especially
since I have another 5 sitting around:

  10 + 5 + 0 + 5

And now I can take advantage of the fact that if I add 0 (the
'additive identity') to something, it makes no difference:

  10 + 5 + 5

We haven't mentioned the associative property yet.  That says that
when we're adding, we can group the additions any way we want.  So we
can do this:

  10 + 5 + 5 = (10 + 5) + 5

             = 15 + 5

             = 20

or this:

  10 + 5 + 5 = 10 + (5 + 5)

             = 10 + 10

             = 20

Either way, we'll get the same answer, but the second one seems easier.

Compare doing it this way to doing the additions one at a time, in the
order they happen to show up:

  6 + 3 + 0 + 2 + 4 + 5
 
      9 + 0 + 2 + 4 + 5

          9 + 2 + 4 + 5

             11 + 4 + 5
 
                 15 + 5

                     20

That wasn't too bad, but in the general case you have to do carries,
and it can really turn into kind of a mess.  If you rearrange and 
group the additions, though, you can make things easier.  

Now, I did this by moving things around, but if you're trying to do it
in your head, you have to keep track of which things you've paired up.
 So I would probably be scanning around with my eyes, matching things
like this:

           ___________
       ___|___        |
      |       |       |
  6 + 3 + 0 + 2 + 4 + 5
  |_______________|

Then mentally, I would just have to add 10, 5, and 5, which is a lot
easier than adding the numbers one at a time. 

Does this make sense?  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Addition

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